A Modified Algorithm for Reducing Calculation Errors in Large Strain Measurement with Strain Gauges

There is no simple linear relationship between strain and potential in strain measurement with strain gauges, especially for large strain measurements. In this paper, a modified algorithm was proposed to improve the accuracy of strain obtained from measured voltage. The strain was calculated from a nonlinear relationship between voltage and strain rather than a linear simplification. Moreover, the corrections for different sensitivity factors of strain gauges and lead wire resistance were considered. The proposed method was suitable for both large and small strain measurements using a quarter bridge, and validated by experimental tests. It is also very easy to be implemented as a software form and used in scientific tests and engineering applications.


Introduction
The strain gauges have widely been used for the small strain measurement in scientific research and engineering applications due to their high precision, low cost and convenience. When large strain measurements are required, optical measurement methods are commonly used. However, on some occasions, for example, for the large strain measurement in a burst test of pressure vessel, strain gauge technology is more suitable because of the high risk and damages.
There are three important things should be considered in using strain gauge technology to measure a large strain [1]. The first is the linearity of strain gauge and adhesion quality. For a small strain measurement, it is a mature technique. Tokyo Measure Institute and other companies have made strain gauge and adhesion suitable for large strain measurement. The second is the linearity of instrument voltage. The linearity within the range of large voltage can be guaranteed by adjusting the amplifier. The third is the relationship between voltage and strain value from the strain gauges. For a small strain measurement, a linear proportional relationship is suitable; for a large strain measurement, using a linear relationship will cause too large an error. In this study, a modified method will be developed to build a nonlinear relationship between the voltage and strain to reduce measurement error.

Basic principle of Strain Gauge Technology
The relationship between strain and resistance change The relationship between strain and resistance change can be expressed as Where k is the sensitivity coefficient of strain gauge, provided by the manufacturer. From Eq. (1), the strain in a structure can be measured by measuring the resistance change of the strain gauge attached to the structure.

Bridge Circuit
where U is the supply voltage.

Standard Strain Gauge Simulator
For a half-bridge circuit, we have Substituting the equations above into Eq.(2) yields Which shows a linear relationship between g U  and R  . A standard strain gauge instrument was made based on this relationship to measure the strain. Quarter Measuring Bridge In engineering applications, a quarter bridge is widely used. We have Substituting the equations above into Eq.(2) yields

Normal Strain Calculation Method
Since / RR  is very small, Equation (4) can be approximately expressed as For an approximation, Eq. (5) is rewritten as

Applied Mechanics and Materials Vols. 13-14 263
The approximation error introduced by using Eq.(6) to calculate When k=2， Eq (7) is expressed as e   (8) Thus, for a strain of 10%, the error introduced by the approximation is 10%. Since Eq. (6) exhibits a linear relationship between the voltage and strain changes and its approximation error is small for small strain measurements, it is widely used.

Modified algorithm of strain calculation
Eq. (4) shows that the relationship between voltage and strain is not linear. In practical applications of large strain measurements, a relationship curve between voltage and strain is calibrated by experiments in order to reduce the calculation error in [2]. This method is not convenient and not adaptable. Therefore, it is more suitable to directly use Eq. (4) rather than the simplified linear relationship between voltage and strain.
From Equation (4), the resistance change rate can be expressed as which can be used to calculate the strain.

Temperature Compensation
In a half bridge, two strain gauges are used: one is used to measure strain, the other is for temperature compensation. The resistance can be expressed as In the room temperature t R  =0，Eqs. (10) and (4) are equivalent.

Sensitivity Correction of Strain Gauge
The sensitivity factor of instrument is different from that of strain gauges, which can be expressed as: Where in k is the sensitivity factor of instrument, in  is the measurement of instrument.

Advances in Experimental Mechanics VI
Thus the strain value from the strain gauge can be written as: Equation (11) can be used to correct the sensitivity factor of different strain gauges to be applicable for the same strain measurement instrument.

Lead wire Resistance Attenuation Correction
The error due to lead wire resistance attenuation can be corrected by using a three-wire circuit, as shown in Figure 2, where strain gauges 1 R and 2 R are used for strain measurement and temperature compensation, respectively. The strain with a lead wire resistance correction is written as Strain Measurements with corrections The strain measurement with temperature compensation and lead wire resistance correction can be obtained by combining Eqs.(11) and (12), expressed as: which presents the corrections of temperature compensation and lead wire resistance corrections, the true strain can be expressed as: The voltage change can be expressed as: Experimental Procedure Strain gauge simulators, SDY 2301 and SDY2306, were used to simulate the behavior of strain. Quarter measuring bridge was used to response the signal from strain gauge simulators, and SDY2102D dynamical strain gauge instrument was used to measure the strain. Quarter measuring bridge could be calculated according to three-wire circuit method as L R is zero. Table 1 shows that the calculation errors of strains obtained from both simplified equation (Eq.(6)) and modified equation (Eq.(17)) are small for a small strain measurement. The error is always small when Eq. (17) is used. However, the error increases with the increasing strain value when Eq. (6) is used: the error increases to more than 1.16% when the strain value reaches 10000  , which can not be neglected; the error increases to 9.12% when the strain value reaches 100000  , which is not acceptable.

Experimental Results
We also did another group experiment by three-wire circuit method and got the same law as Table 1. Lead-wire long is 50 meters and per lead-wire dimension is 0.5mm, we measure lead-wire resistance L R is 2  , R chooses 120  .

Conclusion
In this paper, a modified method was proposed to calculate the strain from the resistance change of the strain gauge, which is suitable for both small and large strain measurements. The corrections for temperature compensation and lead wire resistance attenuation were also considered. The developed method can improve the accuracy of strain measurements and can be used in the scientific research and engineering applications.